Correlation functions in thermal field theory etc Suppose I am studying a field theory at finite temperature or some black hole formation scenario from boundary theory perspective in the sense of AdS/CFT. How is it possible to gain information about them from say looking at the two point functions (propagators) of  operators in field theory? I mean will there be some special pole structures etc. in that Green's function? Is there a generic such behavior? Can you suggest me some references?
 A: I don't have a complete answer but here's something to give you a rough idea.
We consider a Gaussian scalar field with one-point Hamiltonian $\Omega$, i.e. Hamiltonian of this field is given by
$$H = : {1 \over 2} \int {\rm d}^d {\mathbf x} \left( \pi(\mathbf x)^2 + \phi(\mathbf x) \Omega^2\phi(\mathbf x)\right): = \int {\tilde {\rm d} \mathbf p} E(\mathbf p)  a^{\dagger}(\mathbf p)a(\mathbf p)$$
where we introduced $\tilde {\rm d} \mathbf p = {{\rm d} \mathbf p \over (2\pi)^d 2E(\mathbf p)} .$
The Euclidean two-point correlation function of this field is given by
$$\left<{\mathbf x} \right| G_E(t, t') \left|{\mathbf x'} \right> = 
\int {{\rm d}^{d+1} p \over {2 \pi}^{d+1}}
{e^{i p_0(t-t') + {\mathbf p} \cdot ({\mathbf x -\mathbf x'})} \over p_0^2 + E(\mathbf p)^2}$$
while for the thermal Green function we have
$$
\left<{\mathbf x} \right| G_{\beta}(t, t') \left|{\mathbf x'} \right> = 
{1 \over \beta} \sum_{\omega_n = {2 \pi n / \beta}}
\int {{\rm d}^{d} p \over {2 \pi}^{d}}
{e^{i \omega_n(t-t') + {\mathbf p} \cdot ({\mathbf x -\mathbf x'})} \over \omega_n^2 + E(\mathbf p)^2}.
$$
This can be summed and separated into the contributions from the ground state and from the excited states
$$\left<{\mathbf x} \right| G_{\beta}(t, t') \left|{\mathbf x'} \right> = 
\left<{\mathbf x} \right| G_E(t, t') \left|{\mathbf x'} \right> +
\int {{\rm d}^{d} p \over {2 \pi}^{d}}
{1 \over E(\mathbf p)}
{\cosh E(\mathbf p) (t - t') \over e^{\beta E(\mathbf p)} - 1}.$$
Two general observations that can be seen from this:


*

*the thermal Green function has a pole at $\beta E = 2 \pi n$. This comes from the fact that we have compactified time (and so temperature) to a circle.

*in the limit $\beta \to \infty$ the contribution from excited states dies off and we are left with the vacuum Green function.

A: It's a little old now, but try Hiroomi Umezawa, "Advanced Field Theory; Micro, Macro, and Thermal Physics", AIP, 1993.
In a slightly different notation from that used by Marek, the two point function changes from
$$\left<0\right|\phi(x)\phi(x+y)\left|0\right>=\hbar\int 2\pi\delta(k^2-m^2)\theta(k_0)\mathrm{e}^{-ik\cdot y}\frac{\mathrm{d}^4k}{(2\pi)^4},$$
the free Klein-Gordon vacuum state case, to the corresponding thermal state case,
$$\omega_\mathsf{T}\Bigl[\phi(x)\phi(x+y)\Bigr]=\hbar\int 2\pi\delta(k^2-m^2)\theta(k_0)\coth\left[ \!\frac{\hbar k_0}{2\mathsf{k_BT}}\!\right]\mathrm{e}^{-ik\cdot y}\frac{\mathrm{d}^4k}{(2\pi)^4}.$$
On the basis of this presentation, the thermal free field case is no more than a different measure on the forward-light cone, which is, inevitably, not Lorentz invariant (the $k_0$ in the $\coth$ factor picks out a preferred frame). The deformation is no less smooth than the vacuum 2-point function.
Additionally, although the above equations don't show it, the thermal field state is still Gaussian, like the vacuum state, so the whole structure of the thermal state over the free Klein-Gordon field is completely specified by the 2-point function.
One could also construct higher-order deformations of the mass-shell measure, by adding extra factors of $\coth\left[ \!\frac{\hbar k'_0}{2\mathsf{k_BT'}}\!\right]$, or otherwise, quite possibly corresponding to different time-like directions and temperatures.
A presentation of the Klein-Gordon field that is enough to reconstruct the above can be found in my "A succinct presentation of the quantized Klein–Gordon field, and a similar quantum presentation of the classical Klein–Gordon random field", quant-ph/0411156, Phys. Lett. A 338, 8-12(2005), although mostly I'm grinding quite a different axe in that paper.
